Abstract
OpenAI presents a construction of finite-time blowup for the three-dimensional incompressible Navier–Stokes equations, for every positive viscosity. Starting from zero velocity with a smooth force compactly supported in space and time, the proposed flow develops unbounded velocity while its kinetic energy stays uniformly bounded. The paper gives an analytical proof, with an accompanying Lean formalization available separately.
Method
A collapsing vortex is corrected by oscillatory flows whose mean momentum flux cancels the singular residual stress. Further corrections leave a smooth external force through the singular time. The supporting repository contains the Lean formalization and instructions for independent checking.
Limitations
The construction uses an external force. It does not establish finite-time blowup for the unforced Navier–Stokes equations.
AI disclosure
OpenAI reports that an internal multi-agent system produced the proof, followed by Lean formalization and verification using GPT-6 Astra.